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Strassen's theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen's theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen's theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen's theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite-dimensional. [ABSTRACT FROM AUTHOR]

A bounded Hilbert space operator T is hyponormal if T ∗ T − T T ∗ is a positive operator. We consider the hyponormality of Toeplitz operators on a weighted Bergman space. We find a necessary condition for hyponormality in the case of a symbol of the form f + g ¯ where f and g are bounded analytic functions on the unit disk. We then find sufficient conditions when f is a monomial. [ABSTRACT FROM AUTHOR]

We characterize in Hilbert spaces the boundary of the values of maximal monotone operators, by means only of the values at nearby points, which are close enough to the reference point but distinct of it. This allows to write the values of such operators using finite convex combinations of the values at at most two nearby points. We also provide similar characterizations for the normal cone to prox-regular sets. [ABSTRACT FROM AUTHOR]

We consider the null controllability problem fo linear systems of the form y′(t) = Ay(t) + Bu(t) on a Hilbert space Y. We suppose that the control operator B is bounded from the control space U to a larger extrapolation space containing Y. The control u is constrained to lie in a time-varying bounded subset Γ(t)⊂U. From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set Γ(t) contains the origin in its interior at each t>0, the local constrained property turns out to be equivalent to a weighted dual observability inequality of L1 type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets Γ(t) in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that Γ(t) is a closed ball centered at the origin and its radius is time-varying. [ABSTRACT FROM AUTHOR]

In this article, we propose a p-adic analog of complex Hilbert space and consider generalizations of some well-known theorems from functional analysis and the basic study of operators on Hilbert spaces. We compute the K-theory of the analog of the algebra of compact operators and the algebra of all bounded operators. This article contains a survey on results from the thesis of the first author. [ABSTRACT FROM AUTHOR]

The paper deals with semilinear evolution equations in complex Hilbert spaces. Nonlocal associated Cauchy problems are studied and the existence and uniqueness of classical solutions is proved. The controllability is investigated too and the topological structure of the controllable set discussed. The results are applied to nonlinear Schrödinger evolution equations with time dependent potential. Several examples of nonlocal conditions are proposed. The evolution system associated to the linear part is not compact and the theory developed in Okazawa-Yoshii [21] for its study is used. The proofs involve the Schauder-Tychonoff fixed point theorem and no strong compactness is assumed on the nonlinear part. [ABSTRACT FROM AUTHOR]

Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of Chen and Wang [J. Math. Phys. 58(5), 053510 (2017)]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new extragradient-like algorithm for solving a variational inequality problem with a pseudo-monotone and Lipschitz continuous mapping in a Hilbert space. The iterative algorithm combines inertial ideas and hybrid extragradient ideas with the Armijo-like step size rule. Strong convergence of the algorithm is obtained and numerical experiments are provided. [ABSTRACT FROM AUTHOR]

Let (X , Σ , μ) be a finite measure space and consider a Banach function space Y (μ). Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Köthe–Bochner (vector-valued) function spaces. We analyze operator-valued kernels Γ that define integration maps L Γ between Köthe–Bochner spaces of Hilbert-valued functions Y (μ ; 𝒦). We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces L p (g d μ ; 𝒦) and L p ′ (h d μ ; 𝒦)  — where 1 / p + 1 / p ′ = 1 — under the assumption of p -concavity of Y (μ). Equivalently, a new kernel obtained by multiplying Γ by scalar functions can be given in such a way that the kernel operator is defined from L p (μ ; 𝒦) to L p ′ (μ ; 𝒦) in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels. [ABSTRACT FROM AUTHOR]

We study the multivalued mappings on a closed interval of the real line whose values are polyhedra in a separable Hilbert space. The polyhedron space is endowed with the metric of the Mosco convergence of sequences of closed convex sets. A polyhedron is defined as the intersection of finitely many closed half-spaces. The equations of the corresponding hyperplanes involve normals and reals. The normals and reals for a polyhedral multivalued mapping depend on time and are regarded as internal controls. The space of polyhedral multivalued mappings is endowed with the topology of uniform convergence. We study the properties of sets in the space of polyhedral mappings expressed in terms of internal controls. Applying the results, we establish the existence of solutions to polyhedral sweeping processes and study the dependence of solutions on internal controls. We consider minimization problems for integral functionals over the solutions to controlled polyhedral sweeping processes which, along with internal controls, have traditional measurable controls called external. [ABSTRACT FROM AUTHOR]

A Hilbertian (co)algebra is defined as a (co)semigroup object in the monoidal category of Hilbert spaces. The carrier Hilbert space of such an algebra (H , μ) splits as an orthogonal direct sum of its Jacobson radical and the closure of the linear span of a special class of elements, the group-like elements of its adjoint coalgebra (H , μ †) , which by the Riesz representation, correspond to closed maximal modular ideals. When the coproduct is isometric, that is, when μ ° μ † = id , semisimplicity is shown to be equivalent to the existence of adjoints in the sense of Ambrose's H*-algebras. We also prove that the category of semisimple special Hilbertian algebras, that is, semisimple Hilbertian algebras with an isometric coproduct, i.e., essentially the algebras of the form ℓ 2 (X) with the pointwise product, are dually equivalent to a subcategory of pointed sets and base-point preserving maps. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for evolution problems of the form − du/dr(t) ∈ A(t)u(t) + F(t,u(t)) + f(t,u(t)), where, for each t, A(t) : D(A(t)) → 2H, is a maximal monotone operator in a Hilbert space H with continuous, Lipschitz or absolutely continuous variation in time. The perturbation ƒ is separately integrable on [0, T] and separately Lipschitz on H, while F is scalarly measurable and separately scalarly upper semicontinuous on H, with convex and weakly compact values. Several new applications are provided. [ABSTRACT FROM AUTHOR]

Existing viscosity approximation schemes have been extensively investigated to solve equilibrium problems, variational inequalities, and fixed-point problems, and most of which contain that contraction is a self-mapping defined on certain bounded closed convex subset C of Hilbert spaces H for standard viscosity approximation. In particular, if the zero point does not belong to C, the standard viscosity approximation cannot be applied to solve the minimum norm fixed point of some nonlinear operators. In this paper, we introduce three generalized viscosity approximation algorithms with boundary point method for quasi-nonexpansive mappings, which overcome this deficiency above. These three proposed algorithms have simple expressions; moreover, they are easy to implement in the actual computation process. Especially, we can find the minimum norm fixed point of quasi-nonexpansive mappings under contraction is a zero operator. [ABSTRACT FROM AUTHOR]

We prove that measurable multivalued mappings assuming nonempty closed convex values in Hilbert spaces have progressively measurable selectors. We use this assertion to prove the theorem about the existence of martingale solutions to stochastic differential inclusions of the parabolic type in Hilbert spaces. [ABSTRACT FROM AUTHOR]

Let Γ be a countable discrete group, and let π : Γ → GL ⁢ (H) {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group G = H ⋊ π Γ {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group 𝒰 ⁢ (ℓ 2 ⁢ (ℕ)) {\mathcal{U}(\ell^{2}(\mathbb{N}))}) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a II 1 {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space L 0 ⁢ (X , m) {L^{0}(X,m)} of all measurable maps on a probability space. [ABSTRACT FROM AUTHOR]

We consider a quantum channel acting on an infinite-dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence-free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structures are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities. [ABSTRACT FROM AUTHOR]

We study random walks in a Hilbert space H and their applications to representations of solutions to Cauchy problems for differential equations whose initial conditions are number-valued functions on the Hilbert space H. Examples of such representations of solutions to various evolution equations in the case of a finite-dimensional space H are given. Measures on a Hilbert space that are invariant with respect to shifts are considered for constructing such representations in infinite-dimensional Hilbert spaces. According to a theorem of A. Weil, there is no Lebesgue measure on an infinite-dimensional Hilbert space. We study a finitely additive analog of the Lebesgue measure, namely, a nonnegative, finitely additive measure λ defined on the minimal ring of subsets of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles whose products of sides converge absolutely; this measure is invariant with respect to shifts and rotations in the Hilbert space H. We also consider finitely additive analogs of the Lebesgue measure on the spaces lp, 1 ≤ p ≤ ∞, and introduce the Hilbert space H of complex-valued functions on the Hilbert space H that are square integrable with respect to a shift-invariant measure λ. We also obtain representations of solutions to the Cauchy problem for the diffusion equation in the space H and the Schrödinger equation with the coordinate space H by means of iterations of the mathematical expectations of random shift operators in the Hilbert space H . [ABSTRACT FROM AUTHOR]

This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators A and B. The operators A and B are supposed to be densely defined, unbounded, self-adjoint, monotone in the Hilbert space L 2 (Ω) , for some bounded and smooth domain Ω, and have compact resolvents. Our definition of the fractional powers of operators uses the approach via spectral theory. A nonlinearity of double-well type occurs in the phase equation and either a regular or logarithmic potential, as well as a non-differentiable potential involving an indicator function, is admitted in our approach. We show general well-posedness and regularity results, extending the corresponding results that are known for the non-fractional elliptic operators with zero Neumann conditions or other boundary conditions like Dirichlet or Robin ones. Then, we investigate the longtime behavior of the system, by fully characterizing every element of the ω-limit as a stationary solution. In the final part of the paper we study the asymptotic behavior of the system as the parameter σ appearing in the operator B 2 σ that plays in the phase equation decreasingly tends to zero. We can prove convergence to a phase relaxation problem at the limit, in which an additional term containing the projection of the phase variable on the kernel of B appears. [ABSTRACT FROM AUTHOR]

We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution. [ABSTRACT FROM AUTHOR]

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed. [ABSTRACT FROM AUTHOR]

It is shown that each norm closed proper two-sided ideal of a von Neumann algebra is a Lipschitz retract of the algebra. In particular, there exists a Lipschitz retraction from the algebra B(H) of all bounded linear operators on a complex Hilbert space H onto the ideal K(H) consisting of all compact operators. [ABSTRACT FROM AUTHOR]

In a Hilbert space setting, we consider a class of inertial proximal algorithms for nonsmooth convex optimization, with fast convergence properties. They can be obtained by time discretization of inertial gradient dynamics which have been rescaled in time. We will rely specifically on the recent development linking Nesterov's accelerated method with vanishing damping inertial dynamics. Doing so, we obtain a dynamical interpretation of the seminal papers of Güler on the convergence rate of the proximal methods for convex optimization. [ABSTRACT FROM AUTHOR]

Using an explicit catching-up algorithm, we prove the existence of absolutely continuous as well as bounded variation continuous solutions to a second-order perturbed Moreau's sweeping process with the normal cone of a subsmooth moving set, which depends both on the time and on the state. [ABSTRACT FROM AUTHOR]

The Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method does not usually correspond to the calculation of a proximal operator through a closed formula affects the implementability of the algorithm. In this paper, we allow in each block of the objective a further smooth convex function and propose a proximal version of the algorithm, which is achieved by equipping the algorithm with proximal terms induced by variable metrics. For suitable choices of the latter, the solving of the two subproblems in the iterative scheme can be reduced to the computation of proximal operators. We investigate the convergence of the proposed algorithm in a real Hilbert space setting and illustrate its numerical performances on two applications in image processing and machine learning. [ABSTRACT FROM AUTHOR]

We consider the regularization of two proximal point algorithms (PPA) with errors for a maximal monotone operator in a real Hilbert space, previously studied, respectively, by Xu, and by Boikanyo and Morosanu, where they assumed the zero set of the operator to be nonempty. We provide a counterexample showing an error in Xu's theorem, and then we prove its correct extended version by giving a necessary and sufficient condition for the zero set of the operator to be nonempty and showing the strong convergence of the regularized scheme to a zero of the operator. This will give a first affirmative answer to the open question raised by Boikanyo and Morosanu concerning the design of a PPA, where the error sequence tends to zero and a parameter sequence remains bounded. Then, we investigate the second PPA with various new conditions on the parameter sequences and prove similar theorems as above, providing also a second affirmative answer to the open question of Boikanyo and Morosanu. Finally, we present some applications of our new convergence results to optimization and variational inequalities. [ABSTRACT FROM AUTHOR]

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative schemes for finding a common solution of an equilibrium problem and a constrained convex minimization problem. Then, we prove some strong convergence theorems which improve and extend some recent results. [ABSTRACT FROM AUTHOR]

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces. [ABSTRACT FROM AUTHOR]

The interest of this paper is to survey some old and new results on the approximation of fixed points for nonexpansive and pseudocontractive type operators by means of the fastest Krasnoselskij iteration. We consider here generalized pseudocontractions which are also Lipschitzian, a class for which we can use the Krasnoselskij iteration in order to approximate their fixed points. [ABSTRACT FROM AUTHOR]

We prove symplectic non-squeezing for the cubic nonlinear Schrödinger equation on the line via finite-dimensional approximation. [ABSTRACT FROM AUTHOR]

We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest. [ABSTRACT FROM AUTHOR]

We study the heat equation on time‐dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness, and regularity. Of particular interest are properties that characterize the underlying space as a super‐Ricci flow as previously introduced by the second author [51]. Our main result yields the equivalence of dynamic convexity of the Boltzmann entropy on the (time‐dependent) L2‐Wasserstein space,monotonicity of L2‐Kantorovich‐Wasserstein distances under the dual heat flow acting on probability measures (backward in time),gradient estimates for the heat flow acting on functions (forward in time), anda Bochner inequality involving the time‐derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI flow for the (time‐dependent) energy in L2‐Hilbert space and the dual heat flow on probability measures as the unique backward EVI~flow for the (time‐dependent) Boltzmann entropy in L2‐Wasserstein space.© 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given an nc variety V in the nc unit ball Bd, we identify the algebra of bounded analytic functions on V--denoted H∞(V)--as the multiplier algebra MultHV of a certain reproducing kernel Hilbert space HV consisting of nc functions on V. We find that every such algebra H∞(V) is completely isometrically isomorphic to the quotient H∞(Bd)/JV of the algebra of bounded nc holomorphic functions on the ball by the ideal JV of bounded nc holomorphic functions which vanish on V. In order to demonstrate this isomorphism, we prove that the space HV is an nc complete Pick space (a fact recently proved--by other methods--by Ball, Marx, and Vinnikov). We investigate the problem of when two algebras H∞(V) and H∞(W) are (completely) isometrically isomorphic. If the variety W is the image of V under an nc analytic automorphism of Bd, then H∞(V) and H∞(W) are completely isometrically isomorphic. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case d = ∞ we need to assume that the isomorphism is also weak-* continuous). We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of Bd and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case d = ∞ without further assumptions. Along the way, we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases. [ABSTRACT FROM AUTHOR]

We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras to s-finite von Neumann algebras (that is, von Neumann algebras with a faithful state, which includes those on a separable Hilbert space or with separable predual). To achieve this extension, completely new strategies had to be invented at certain key points, ultimately resulting in a more operator algebraic proof of the result. Ueda showed in the case of finite von Neumann algebras that his peak set theorem is the fountainhead of many other very elegant results, like the uniqueness of the predual of such subalgebras, a highly refined F & M Riesz type theorem, and a Gleason-Whitney theorem. The same is true in our more general setting, and indeed we obtain a quite strong variant of the last mentioned theorem. We also show that set theoretic issues dash hopes for extending the theorem to some other large general classes of von Neumann algebras, for example finite or semi-finite ones. Indeed certain cases of Ueda's peak set theorem for a von Neumann algebra M may be seen as 'set theoretic statements' about M that require the sets to not be 'too large'. [ABSTRACT FROM AUTHOR]

In this article, a projection-type method for mixed variational inequalities is proposed in Hilbert spaces. The proposed method has the following nice features: (i) The algorithm is well defined whether the solution set of the problem is nonempty or not, under some mild assumptions; (ii) If the solution set is nonempty, then the sequence generated by the method is strongly convergent to the solution, which is closest to the initial point; (iii) The existence of the solutions to variational inequalities can be verified through the behavior of the generated sequence. The results presented in this article generalize and improve some known results. [ABSTRACT FROM AUTHOR]

Kernel mean embeddings have become a popular tool in machine learning. They map probability measures to functions in a reproducing kernel Hilbert space. The distance between two mapped measures defines a semi-distance over the probability measures known as the maximum mean discrepancy (MMD). Its properties depend on the underlying kernel and have been linked to three fundamental concepts of the kernel literature: universal, characteristic and strictly positive definite kernels. The contributions of this paper are three-fold. First, by slightly extending the usual definitions of universal, characteristic and strictly positive definite kernels, we show that these three concepts are essentially equivalent. Second, we give the first complete characterization of those kernels whose associated MMD-distance metrizes the weak convergence of probability measures. Third, we show that kernel mean embeddings can be extended from probability measures to generalized measures called Schwartz-distributions and analyze a few properties of these distribution embeddings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid iterative method to approximate a common solution of split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. We prove that sequences generated by the proposed hybrid iterative method converge strongly to a common solution of these problems. Further, we discuss some applications of the main result. We also discuss a numerical example to demonstrate the applicability of the iterative method. The method and results presented in this paper extend and unify the corresponding known results in this area. [ABSTRACT FROM AUTHOR]

In a Hilbert space, we analyze the convergence properties of a general class of inertial forward-backward algorithms in the presence of perturbations, approximations, errors. These splitting algorithms aim to solve, by rapid methods, structured convex minimization problems. The function to be minimized is the sum of a continuously differentiable convex function whose gradient is Lipschitz continuous and a proper lower semicontinuous convex function. The algorithms involve a general sequence of positive extrapolation coefficients that reflect the inertial effect and a sequence in the Hilbert space that takes into account the presence of perturbations. We obtain convergence rates for values and convergence of the iterates under conditions involving the extrapolation and perturbation sequences jointly. This extends the recent work of Attouch-Cabot which was devoted to the unperturbed case. Next, we consider the introduction into the algorithms of a Tikhonov regularization term with vanishing coefficient. In this case, when the regularization coefficient does not tend too rapidly to zero, we obtain strong ergodic convergence of the iterates to the minimum norm solution. Taking a general sequence of extrapolation coefficients makes it possible to cover a wide range of accelerated methods. In this way, we show in a unifying way the robustness of these algorithms. [ABSTRACT FROM AUTHOR]

A tripartite state ρABC forms a Markov chain if there exists a recovery map RB→BC acting only on the B-part that perfectly reconstructs ρABC from ρAB. To achieve an approximate reconstruction, it suffices that the conditional mutual information I(A:C|B)ρ is small, as shown recently. Here we ask what conditions are necessary for approximate state reconstruction. This is answered by a lower bound on the relative entropy between ρABC and the recovered state RB→BC(ρAB). The bound consists of the conditional mutual information and an entropic correction term that quantifies the disturbance of the B-part by the recovery map. [ABSTRACT FROM AUTHOR]

We study a version of the generalized (h, ϕ)-entropies, introduced by Salicrú et al. [M. Salicrú et al., Commun. Stat. Theory Method. 22, 2015 (1993)], for a wide family of probabilistic models that includes quantum and classical statistical theories as particular cases. We extend previous works by exploring how to define (h, ϕ)-entropies in infinite dimensional models. [ABSTRACT FROM AUTHOR]

Weak supercyclicity is related to weak stability, which leads to the question that asks whether every weakly supercyclic power bounded operator is weakly stable. This is approached here by investigating weak l-sequential supercyclicity for Hilbert-space contractions through Nagy–Foliaş–Langer decomposition, thus reducing the problem to the quest of conditions for a weakly l-sequentially supercyclic unitary operator to be weakly stable, and this is done in light of boundedly spaced subsequences. [ABSTRACT FROM AUTHOR]

In this paper, we consider the split equality common fixed point problem of infinite families of demicontractive mappings in Hilbert spaces. We introduce a simultaneous iterative algorithm for solving the split equality common fixed point problem of infinite families of demicontractive mappings and prove a strong convergence of the proposed algorithm under some control conditions. [ABSTRACT FROM AUTHOR]

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's KSr Conjecture (2004) in the form of approximate Lyapunov's theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov's theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem. [ABSTRACT FROM AUTHOR]

We study operators defined on a Hilbert space defined by a self-affine Delone set Λ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain lim sup law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or Rd-invariant distributions of a dynamical system defined by Λ. We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on ℓ2(Λ) and show that the errors of convergence in Shubin’s formula are given by these traces. [ABSTRACT FROM AUTHOR]

Abstract: Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras. [ABSTRACT FROM AUTHOR]

Establishing the convergence of splines can be cast as a variational problem which is amenable to a Γ-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, n, as λn=n-p. Using standard theorems from the Γ-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for p≤12. Without further assumptions, we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose p>12. [ABSTRACT FROM AUTHOR]

We modify a hybrid method and a proximal point algorithm to iteratively find a zero point of the sum of two monotone operators and fixed point of nonspreading multivalued mappings in a Hilbert space by using the technique of forward-backward splitting method. The strong convergence theorem is established and the illustrative numerical example is presented on this work. The results of this paper extend and improve some well-known results in the literature. [ABSTRACT FROM AUTHOR]

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ℋ, we study three cases of measures. We first show that, for ℋ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ℋ. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures. [ABSTRACT FROM AUTHOR]

This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3, 2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned by them using appropriate topologies on our spaces. Finally, we discuss the properties of the functionals that form the continuous basis. [ABSTRACT FROM AUTHOR]

In this work, we develop the necessary conditions of optimality for partially observed control problems governed by stochastic integrodifferential equations of neutral type on Hilbert spaces driven by relaxed controls. Under suitable conditions, we establish the existence and the uniqueness of mild solution which has a continuous modification. We prove the continuity of the control to the solution map. Then we consider an optimal control problem of Bolza type. Using these results we give necessary conditions of optimality for a class of stochastic integrodifferential equations of neutral type on Hilbert spaces. [ABSTRACT FROM AUTHOR]